**3. Wave-mixed bioreactor**

The rocker-type motion of a wave-mixed bioreactor can be described by a harmonic oscillation function [18]. This allows the deflection angle *φ* at time point *t* to be predicted by eq. (5), whereby *φmax* corresponds to the maximum deflection angle and *ω* to the angular velocity, which is calculated using the frequency *f* (eq. 6). The frequency itself corresponds to the set rocking rate:

$$
\rho\_t = \rho\_{\text{max}} \cdot \sin(\overline{a}t) \tag{5}
$$

$$
\overline{w} = 2\pi \cdot f \tag{6}
$$

The motion can be realised with the help of a changing gravity vector *g* ! (9*:*81 m s�2) that is decomposed into its *x* and *y* components for a given deflection angle (**Figure 4b**). Thus, the gravity vector *gx* and *gy* can be calculated using eqs (7) and (8). Choosing a maximum deflection angle of 10<sup>∘</sup> and a rocking rate of 25 rpm leads to the outcomes in (**Figure 4c** and **d**):

$$
\overrightarrow{\mathbf{g}}\_x = \overrightarrow{\mathbf{g}} \cdot \sin \phi\_t \tag{7}
$$

$$\overrightarrow{\mathbf{g}}\_{\mathcal{Y}} = \overrightarrow{\mathbf{g}} \cdot \cos \boldsymbol{\varrho}\_{t} \tag{8}$$

Another way to mimic the movement of the bag is to rotate or translate the mesh with a user-defined function according to eq. (9):

#### **Figure 4.**

*Representation of a) the 3D-scanned CAD geometry of the Flexsafe RM 2 L basic, b) the decomposition of the gravity vector into its x and y components at a deflection of φ, c) and d) the decomposition of the gravity vector into its x and y components over time at a maximum deflection of 10*<sup>∘</sup> *and 25 rpm.*

$$
\overline{\alpha}\_{l} = \overline{\alpha} \left( \rho\_{\max} \cdot \frac{\pi}{180} \right) \cdot \cos(\overline{\alpha}t) \tag{9}
$$

Both of the above-mentioned methods for implementing the motion in the wave-mixed bioreactor were used for simulations of a two-phase air and water system in a Flexsafe RM 2 L basic single-use bioreactor bag from Sartorius AG (see used CAD geometry in **Figure 4a**). The simulation was performed transiently for liquid volumes of 0.5 L, 1 L, and 1.5 L in Ansys Fluent using velocity-pressure coupling, the SIMPLE algorithm, and the volume of fluid (VOF) model of Hirt and Nichols [19] with two Eulerian phases and a mesh consisting of 1*:*<sup>5</sup> � <sup>10</sup><sup>6</sup> tetrahedrons. A no-slip assumption was chosen for the walls of the Flexsafe RM. Turbulence modelling was performed using the *k*-*ω* model. To achieve sufficient convergence for this procedure, time steps of 10�<sup>4</sup> s were chosen at the beginning of the simulation.

The CFD simulations were validated by comparing experimentally determined liquid level (**Figures 5** and **6**) and PIV measurements (**Figure 7**) to the simulations. This showed good qualitative and quantitative congruence of the geometric expression of the wave along the bag cross section for different volumes, deflection angles, and rocking rates. The relative deviations remained in the order of magnitude of less than 10% for all of the simulations, which can be attributed to a slight temporal offset of the images being compared. This resulted from the fact that, due to

#### **Figure 5.**

*Qualitative comparison of the fluid surfaces from the fluid level measurements (top) and the numerical simulations (bottom) at a) a fluid volume of 0.5 L, a maximum deflection angle of 6*<sup>∘</sup> *at 35 rpm, b) a fluid volume of 1 L, a maximum deflection angle of 6*<sup>∘</sup> *at 25 rpm, and c) a fluid volume of 1.5 L, a maximum deflection angle of 10*<sup>∘</sup> *at 25 rpm; all images shown have an instantaneous deflection of 3*<sup>∘</sup> *.*

#### **Figure 6.**

*Dimensionless comparison of the fluid surfaces from the level measurements and the numerical simulations with different motion implementations for a fluid volume of 1 L, a maximum deflection angle of 6*<sup>∘</sup> *and an instantaneous deflection of 4*<sup>∘</sup> *at 25 rpm; simulation 1 corresponds to the motion implementation by means of decomposition of the gravity vector into its x and y components; simulation 2 was realised by means of the mesh motion.*

the large amounts of data generated during the numerical simulations, the simulation results were only saved every hundredth of a second. As a result, minimal deviations can occur in the comparisons of the wave characteristics at each full degree, because results might not always be available for the exact time point. Another cause of slight deviations is the use of a rigid, 3D-scanned bag shape for the simulations (**Figure 4a**), since in reality slight changes in shape can be observed and, depending on how the bag

#### **Figure 7.**

*Qualitative comparison of the fluid velocities from the PIV measurements (top right) and the results of the CFD simulation (top left) for 1 L at 10*<sup>∘</sup> *and 25 rpm and a quantitative comparison (bottom) of the velocities along the introduced line with a distance of 10 mm above the ground at an instantaneous deflection of 5*<sup>∘</sup> *; the respective legends indicate the fluid velocities from 0.0 m s*<sup>1</sup> *to 0.6 m s*<sup>1</sup>*.*

is fixed, small folds may also form, neither of which were taken into account in the simulation [20].

When comparing the results of the different motion implementations, it becomes clear that only marginal differences occurred between the two simulations (**Figure 6**). The relative deviations of the simulations to each other amounted to a maximum of 5%, which can also be attributed to the previously mentioned reasons as well as to the different mesh handling of Ansys Fluent, since, for example, the change of the mesh position in the Cartesian coordinate system must be taken into account [21]. Nevertheless, this way of handling the mesh appears to result in significantly shorter computation time when opting for the mesh motion option. Computation times of at least 12 weeks were required to reach quasi-stationary periods for gravity vector decomposition, while the computation time for mesh motion was only 4 weeks. Therefore, the latter should be considered the method of choice for future simulations. Validation by means of PIV measurements at specific deflection angles also proved successful. The planes from the PIV measurements were compared to the same planes from the simulations, and the velocity profiles along introduced lines were quantitatively compared. Consequently, good agreement can be assumed to indicate that the simulation results are of high quality. Based on the results, the periodically changing power input and the mean volume-averaged shear stresses could be determined (example shown in *Computational Fluid Dynamics for Advanced Characterisation of Bioreactors Used… DOI: http://dx.doi.org/10.5772/intechopen.109849*

#### **Figure 8.**

*Characterisation of the specific power input of the numerical simulations at 1 L, 10*<sup>∘</sup> *, and 25 rpm over an entire period (top) as well as representation of the maximum values of the mean volume-averaged shear stresses over an interval of one period (bottom).*

**Figure 8**), which with values of up to 250 W m<sup>3</sup> and a maximum of 0*:*026 N m<sup>2</sup> indicate good conditions for the cultivation of mammalian cells. For this reason, it comes as no surprise that they are most commonly used for mammalian cell inoculum production processes in feeding and, more recently, perfusion mode [22–25].
